The slope of fibred surfaces: unitary rank and Clifford index
TThe slope of fibred surfaces: unitary rank and Cliffordindex
Enea Riva, Lidia Stoppino ∗ February 9, 2021
Abstract
We prove new slope inequalities for relatively minimal fibred surfaces, showingan influence of the relative irregularity q f , of the unitary rank u f and of the Cliffordindex c f on the slope. The argument uses Xiao’s method and a new Clifford-typeinequality for subcanonical systems on non-hyperelliptic curves. Let f : S → B be a relatively minimal fibred surface. Let K f = K S − f ∗ K B beits relative canonical divisor and χ f = χ ( O S ) − χ ( O F ) χ ( O B ) be its relative Eulercharacteristic (see Section 3.1). A slope inequality for the fibred surface is an inequalityof the form: K f ≥ aχ f , (1.1)where a > K f ≥ g − g χ f , (1.2)where g = g ( F ), the genus of a general fibre F . A third proof was given later byMoriwaki in [31]. So, here a is an increasing function of g .The rank g vector bundle f ∗ ω f over the base curve B is called the Hodge bundleof the fibred surface. Note that by Leray’s spectral sequence χ f coincides with thedegree of the Hodge bundle deg f ∗ ω f , and the slope inequality (1.2) can be rephrasedas follows: K f ≥ g − µ ( f ∗ ω f ) = 2 deg( ω F ) µ ( f ∗ ω f ) = 2( K f F ) µ ( f ∗ ω f ) , where µ ( E ) as usual denotes Mumford’s slope of a vector bundle E (see Section 3.2).After the seminal papers cited above, several results have been obtained by manyauthors proving an influence of other natural geometric invariants of the fibred surfaceon this inequalities. Call b := g ( B ) the genus of the base curve. Let us consider inparticular the following invariants. (see also Section 3.1): ∗ The authors were partially supported MIUR PRIN 2017 Moduli spaces and Lie Theory , by MIUR,Programma Dipartimenti di Eccellenza (2018-2022) - Dipartimento di Matematica F. Casorati, Uni-versit`a degli Studi di Pavia and by INdAM (GNSAGA) a r X i v : . [ m a t h . AG ] F e b The relative irregularity q f = q ( S ) − b .- The unitary rank u f ([24]), i.e. the rank of the unitary summand U in the secondFujita decomposition of the Hodge bundle.- The gonality of a general fibre gon( f ) = gon( F ).- The Clifford index of a general fibre c f = Cliff( F ).In general, q f ≤ g and equality holds if and only if the fibration is trivial. Many resultsare known about the relations between q f , u f and c f . We recall in particular that fornon-isotrivial fibred surfaces, we have q f ≤ g + 1 if b = 0 ([41]), but there exist fibredsurfaces such that q f > g + 1 [35]. Moreover, we have ([40, 16]) u f ≤ g +16 and ([7, 24]) q f ≤ u f ≤ g − c f . (1.3)A sharp bound is not known, it is predicted by the modified Xiao conjecture [7, 24].The first two invariants satisfy inequality u f ≥ q f (see Section 3.2), but there arefibred surfaces where strict inequality hold: thanks to the results of Catanese andDettweiler ([13] and [14]) we know that these fibred surfaces are -modulo base change-precisely the ones having unitary summand U with infinite monodromy (see [24] andSection 3.1, Remark 3.8).The relative irregularity has a clear geometric meaning as the dimension of a fixedabelian variety which is the image of the Jacobians of all the smooth fibres via thehomomorphism induced by the inclusion [11]. On the other hand, the unitary rank hasa more elusive meaning (see [24], [23]).The question of an increasing bound depending on q f dates back to the originalpaper of Xiao, where he proves that, if q f >
0, then K f ≥ χ f and, moreover, that if K f = 4 χ f , then q f = 1 ([40, Cor.2, Thm.3]).Let us consider the gonality and the Clifford index of the fibration: recall that(Section 2.1) gon( f ) − ≤ c f ≤ gon( f ) − . It was proved in the semistable case by Cornalba and Harris (and generalized by thesecond author [36, Prop. 2.4]) that if a non-locally trivial fibred surface satisfies equal-ity in (1.2), then necessarily it is hyperelliptic (i.e. with minimal gonality 2 i.e. Cliffordindex 0). Thus, one naturally would expect that there exists a function in (1.1) in-creasing with gon( f ), with some genericity assumption needed, as observed by Barjaand the second author in [9, Remark 3.6].Some results are known for small gonality ([25, 9, 12, 18, 19]). A stunning approachusing relative Koszul sequences is proposed in [26], proving in particular that for odd g and general gonality (cid:98) g +32 (cid:99) = g +32 , hence Clifford index (cid:98) g − (cid:99) = g − , we have: K f ≥ g − g + 1 χ f . (1.4)Another step towards an answer to both problems was given by Barja and the secondnamed author in [8] with the following result: K f ≥ g − g − (cid:98) m/ (cid:99) χ f , (1.5)2here m = min { q f , c f } . This bound is interesting if both invariants are big with respectto g : this can very well happen, as proved in loc. cit. by providing several examples.Very recently, Lu and Zuo introduced a yet another very natural technique, usingthe relative multiplication map Sym f ∗ ω f −→ f ∗ ω ⊗ f , combined with Xiao’s method. Thanks to this technique, the two authors were ableto improve [8] in both directions. Firstly, they obtain an inequality with a = a ( g, q f )increasing with the relative irregularity [27]: K f ≥ g − g − q f / χ f . (1.6)Moreover, they proved in [28] the following: if a general fibre of f is general in the k -gonal locus D k in M g and g ≥ ( k − , then K f ≥ (5 k − g − k − g + 2) χ f . (1.7)No bound is known -to our knowledge- involving the unitary rank u f .In this paper we prove new bounds depending increasingly on c f , q f and u f . Letus summarize here the main results obtained. Theorem (Theorems 4.1 and 4.2) . Let f : S → B be a relatively minimal fibred surfaceof genus g ≥ ; let m := min { q f , c f } . The following inequalities hold: K f ≥ g − − mg − m χ f , (1.8) K f ≥ (2 g − − u f )( g − u f ) χ f if u f ≤ c f ; (2 g − − c f )( g − − u f )( g − − c f )( g − u f ) χ f if u f ≥ c f . (1.9) Remark 1.10.
Let us compare our results with the known results. • The first inequality (1.8) improves (1.5) and, more importantly, is greater than(1.6) in case q f ≤ c f . On the other hand in case q f ≥ c f , inequality (1.8) givesa bound increasing with the Clifford index. Inequality (1.7) can be better, butinequality (1.8) holds also when (1.7) is not applicable: no genericity assumptionsis needed, nor assumptions on g (cid:29) m . Moreover, for m big, or u f and c f closeto g − , the bound of inequalities (1.8) and (1.9) becomes close to 6 (see Remark1.11 below). • Inequalities (1.9) are the first known slope inequalities showing an influence of u f . • Inequalities (1.9) are of particular interest in view of the fact cited above that u f can be strictly bigger than q f . In Section 5, following [15], we give a first exampleof a fibred surface where the second inequality is new. This fibred surface hasinvariants g = 6, q f = 0, c f = u f = 2, and is not bielliptic. The bound of (1.9) is K S ≥ χ f , while the other previously known bounds are strictly smaller or notapplicable. 3 emark 1.11. Note that all our bounds are asymptotically close to 4 for g (cid:29)
0, andthis is natural in view of all the known examples and conjectures. But when m is bigwith respect to g , the slope gets bigger, going asymptotically to 6. Let us observe thatfor odd genus, if the Clifford index is maximal Cliff( f ) = (cid:98) g − (cid:99) and if q f ≥ g − (1.8)becomes Konno’s bound (1.4). For Clifford index (hence gonality) close to g − , yet notmaximal, these bounds are new.Our arguments make use of Xiao’s method (Section 3.2). Basically, Xiao’s techniqueworks as follows: given a subsheaf of the Hodge bundle G ⊆ f ∗ ω f , consider the linearsub-canonical system G ⊗ C ( t ) ⊆ H ( F, K F ) induced on a general fibre F = f ∗ ( t ). Ifone has a lower lower estimate on the ratio of degree over projective dimension of thelinear subsystems of G ⊗ C ( t ), then the method produces an inequality of the form K f ≥ b deg( G ), where b is a positive number depending on the lower estimate above.See Section 3.2 and Theorem 3.20 for precise statements. Taking as G the whole Hodgebundle, Clifford’s Theorem (see Section 2.1) gives the slope inequality (1.2).It is thus very natural to try and apply Xiao’s method to the ample summand A of the second Fujita decomposition of the Hodge bundle (3.6), as deg A = χ f . In[10], [8] the analog approach is discussed with the positive summand of the first Fujitadecomposition (3.5). One of the difficulties with these approaches is that there seemsto be no control on the base locus of the linear sub-canonical systems induced by A onthe general fibres of f , neither on the linear stability (ref. Section 2.1) of this system.However, one can still look to a lower bound for the ratio of degree over projectivedimension of the linear subcanonical systems that improves Clifford’s bound 2.This is what we do in our paper, obtaining a new Clifford-type inequality for sub-canonical systems over a non-hyperelliptic curve C , only depending on the codimensionand on the Clifford index of C . This gives also the desired control on the base locus ofthe subcanonical systems. Theorem (Theorem 2.13) . Let C ⊆ P g − be a canonical non-hyperelliptic curve. Let V ⊆ H ( C, ω C ) be a linear subspace of codimension k ≤ g − . Then for any W ⊆ V subspace of dimension dim W ≥ , we have: deg | W | dim | W | ≥ g − − mg − m − , where m := min { k, Cliff( C ) } . Although the motivation in this paper is to apply Xiao’s technique, we believe thatthis result is interesting on its own. The arguments are of genuine geometric classicalflavour.The above result implies a stability result, as follows (see Section 2.1 for the defini-tions).
Corollary 1.12 (Corollary 2.17) . Given V ⊆ H ( C, ω C ) a vector subspace of codimen-sion k and dimension ≥ , with k ≤ Cliff( C ) . Then deg | V | ≥ g − − k , i.e. the baselocus of | V | has degree smaller or equal to k . If deg | V | = 2 g − − k , then | V | is linearlysemistable and in particular it is Chow semistable. emark 1.13. This result should be compared also to [30], where linear stability oflinear systems on curves is discussed in relation to the Clifford index.
Remark 1.14.
The slope inequalities have applications both to the geography of sur-faces of general type (see for instance [34]) and to the ample cone of the moduli spaceof curves (see for instance [31] and [22]). These perspectives were the original point ofview of Xiao and of Cornalba and Harris respectively. In the last years, many authorshave treated the case of slope inequalities of fibrations over curves with total space ofhigher dimension (see for instance [6]).The paper is organized as follows. In Section 2, after some preliminaries on canonicalcurves and linear stability, we prove the main Clifford-type result for non-completesub-canonical systems on non-hyperelliptic curves. We then discuss some stabilityconsequence and give some natural examples.In Section 3 we start by reviewing in 3.1 some basic results on fibred surfaces andtheir relative invariants. Then in 3.2 and 3.3 we give a review of the main theorems ofXiao’s technique, in the form needed for our arguments. We state Xiao’s method forfibred surfaces in full generality, following Konno’s and Barja’s papers, for any locallyfree subsheaf G of f ∗ O S ( D ), where D is a nef divisor on S .The proof of the main inequalities is carried on in Section 4.In the last Section, following Catanese and Dettweiler’s examples, we provide a firstexample of a fibred surface such that the inequality in the main theorem involving u f is new. Notation 1.15.
We work over the complex field C . All varieties, unless otherwisestated, are assumed to be smooth and projective. Given a variety X and a divisor D on X , H ( X, D ) means as usual H ( X, O X ( D )). Acknowledgements
We wish to thank Miguel `Angel Barja for countless invaluable conversations on thesubject, and for a careful revision of a first draft of the paper.
Let C be a smooth projective curve of genus g ( C ) = g ≥
2, and let K C (resp. ω C ) itscanonical divisor (resp. line bundle). Let φ K : C −→ P ( H ( C, ω C ) ∨ ) ∼ = P g − be its canonical morphism. Assume that C is non-hyperelliptic, i.e that φ K is anembedding. Often, with abuse of notation, we identify C and its points with thecorresponding canonical image.Given a linear subspace V ⊆ H ( C, ω C ), let us consider: Ann ( V ) := { θ ∈ H ( C, ω C ) ∨ | θ ( v ) = 0 ∀ v ∈ V } ⊆ H ( C, ω C ) ∨ .
5e call this subspace annihilator of V . Let A nn( V ) = P ( Ann ( V )) ⊆ P ( H ( C, ω C ) ∨ )be its projectivisation. Observe that the dimension of A nn( V ) is the codimension of V minus one. Definition 2.1.
Given an effective divisor D on C , its projective span isspan( D ) = span φ K ( D ) := A nn( H ( C, ω C ( − D )) ⊆ P ( H ( C, ω C ) ∨ ) Example 2.2.
Given a point p ∈ C , span φ K ( p ) = { p } , while span φ K (2 p ) is the linetangent to C in P g − , span φ K (3 p ) is the osculating plane to C , and so on. For n distinct points p , . . . , p n on C if we call D = p + . . . + p n , we have that span φ K ( D )coincides with the linear projective span of the points in P g − . Theorem 2.3 (Geometric version of Riemann-Roch [1]) . Given an effective divisor D of degree d on a smooth non-hyperelliptic curve C of genus g ≥ , we have: dim(span( D )) = dim(span φ K ( D )) = d − − dim | D | = d − h ( C, D ) . Given a linear subspace V ⊆ H ( C, ω C ) we call the base locus D V of the linearsystem | V | the scheme-theoretic intersection D V := A nn( V ) ∩ C. Observe that theevaluation map of V is surjective onto ω C ( − D V ). Definition 2.4. (Gonality) The gonality gon( C ) of C is the following integer:gon( C ) := min { deg( π ) | π : C → P is a surjective morphism } = min { m | ∃ g m over C } . Definition 2.5. (Clifford index) Given a curve C of genus g ≥
4, we define its Cliffordindex Cliff( C ) as:Cliff( C ) := min { deg( D ) − | D | ) | h ( C, D ) ≥ , h ( C, D ) ≥ } . In case g = 2 , • if g = 2, Cliff( C ) := 0; • if g = 3, Cliff( C ) := 0 (resp. 1) if C is hyperelliptic (resp. trigonal).For every divisor D such that h ( C, D ) ≥ h ( C, D ) ≥
2, we say that D contributes to the Clifford index. Remark 2.6.
Clifford’s Theorem ([1, pp.107-108]) is equivalent to the following state-ment: for any curve C of genus g ≥
2, Cliff( C ) ≥ C is hyperelliptic.Gonality and Clifford index are well studied invariants, we briefly recall some clas-sical results about them. We have the following upper bounds:gon( C ) ≤ (cid:98) g + 32 (cid:99) , Cliff( C ) ≤ (cid:98) g − (cid:99) , with equality holding for a general curve in M g . Gonality also has a very naturalgeometric interpretation via Geometric Riemann-Roch Theorem:6 roposition 2.7. For every effective divisor D over C , dim(span( D )) ≤ deg( D ) − . If dim span( D ) < deg( D ) − , then deg D ≥ gon( C ) . If on the other hand k is aninteger greater or equal to gon( C ) , then there exists a divisor D of degree deg D = k with dim span D < deg D − .Proof. The first inequality is straightforward from Geometric Riemann Roch 2.3. Sup-pose now that dim span( D ) < deg( D ) −
1; by Riemann Roch again h ( C, D ) ≥
2. Sothere exists a linear subspace V ⊆ H ( C, D ) of dimension 2 and degree ≤ deg D . Thusgon( C ) ≤ deg D . The other implication is immediate. Remark 2.8.
For example, a non-hyperelliptic curve C is trigonal if and only if thereexist three collinear points on C , a curve C is 4-gonal (i.e. gon( C ) = 4) if and only ifevery three points p , p , p of C are not collinear, but there exist a 4-uple of points of C that spans a plane. Remark 2.9.
Ballico [4] proved thatgon( C ) − ≤ Cliff( C ) ≤ gon( C ) − . (2.10)and for a general curve C in the locally closed subset of curves in the moduli space ofgonality gon( C ), it holds equality on the right.Eventually, we recall the following definition due to Mumford [32]. Definition 2.11. (Linear (semi)stability) A linear system | V | over C is linearly stable (resp. semistable) if for every linear subsystem | W | ⊆ | V | we have:deg | W | dim | W | > deg | V | dim | V | (resp. ≥ ) Remark 2.12.
Let us make some remarks. • The linear system | V | and its linear subsystems | W | are not necessarily complete; • If | V | ⊆ | L | has a non zero base locus D V , then the linear subsystem: V ( − D V ) := V ∩ H ( C, L − D V )destabilizes it, because deg | V ( − D V ) | < deg | V | but dim | V ( − D V ) | = dim | V | .So, systems with base points are linearly unstable. • Again, Clifford’s theorem can be rephrased saying that the canonical system ona curve is linearly semistable, and it is stable if and only if the curve is non-hyperelliptic. • Linear stability was introduced by Mumford in order to develop a simple methodto prove GIT stability results, indeed, it is proven in [32] that linear semistabilityimplies Chow stability and in [2] that linear stability implies Hilbert stability.7 .2 The main result
A linear system | V | is linearly stable if its ratio deg | V | / dim | V | bounds from belowthe ratio d/r for any g rd ⊆ | V | . Changing point of view, given a linear system on acurve, one can ask for a lower bound for this ratio d/r possibly lower than the originalratio deg | V | / dim | V | . This is what we do for canonical subsystems of non-hyperellipticcurves, obtaining a bound depending on the codimension and on the Clifford index ofthe curve. Theorem 2.13.
Let C ⊆ P g − be a canonical non-hyperelliptic curve. Let V ⊆ H ( C, ω C ) a linear subspace of codimension k ≤ g − . Then for any W ⊆ V sub-space of dimension dim W ≥ , we have: deg | W | dim | W | ≥ g − − mg − m − . where m := min { k, Cliff( C ) } .Proof. For any W ⊆ V we have the evaluation morphism: W ⊗ O C (cid:16) ω C ( − D W ) , where D W := A nn( W ) ∩ C is the base locus of | W | .We begin by considering the case m = k . Lemma 2.14. If k ≤ Cliff( C ) , then deg | V | = deg( ω C ( − D V )) ≥ g − − k , i.e. deg D V ≤ k .Proof. We split the proof of the lemma in two cases: • If h ( C, D V ) ≥
2, since h ( C, ω C ( − D V )) ≥ dim V ≥
2, then both D V and ω C ( − D V ) contributes to the Clifford index of C , so we have:deg( ω C ( − D V )) ≥ h ( C, ω C ( − D V )) − C ) ≥ g − k − k = 2 g − − k, as wanted. • If h ( C, D V ) = 1, by the geometric version of Riemann-Roch (Theorem 2.3), wehave that: dim(span( D V )) = deg D V − h ( C, D V ) = deg D V − . Now, span( D V ) ⊆ A nn V by construction, anddim A nn( V ) = g − − dim( V ) = g − − ( g − k ) = k − . Therefore, we can conclude that deg D V ≤ k , and the claim is proven.Let’s go back to the proof of Theorem 2.13. Let W (cid:40) V , with dim W ≥
2. As donefor Lemma 2.14, we analyze the two following cases:8i) If h ( C, D W ) ≥
2, hence D W contributes to Cliff( C ) since h ( C, D W ) = h ( C, ω C ( − D W )) ≥ dim W ≥
2, then:deg ω C ( − D W ) ≥ h ( C, ω C ( − D W )) −
1) + Cliff( C ) ≥ W −
1) + k. Hence:deg | W | dim | W | = deg ω C ( − D W )dim | W | ≥ k dim | W | ≥ k dim | V | = 2 g − − kg − k − , as wanted.(ii) If h ( C, D W ) = 1 we can conclude deg D W ≤ dim( A nn W ) + 1 as in the proof oflemma 2.14. Setting k W := dim( A nn W ) + 1 = codim( W ), we have:deg | W | dim | W | = 2 g − − deg D W g − k W − ≥ g − − k W g − k W − . Since W ⊆ V we can conclude that k W ≥ k .Now, consider the function: f : [0 , g − → R f ( t ) := 2 g − − tg − t − . (2.15)As f (cid:48) ( t ) = g − g − t − > ∀ t ∈ [0 , g − , we have that f is monotone strictly increasing. So, since k W ≥ k , we obtain:deg | W | dim | W | ≥ f ( k W ) ≥ f ( k ) = 2 g − − kg − k − , as wanted.Let us now treat the case k ≥ Cliff( C ) =: c . We prove that for any W ⊆ V , withdim W ≥
2: deg | W | dim | W | ≥ g − − cg − c − . Like we did above, we focus on two cases:(i) if h ( C, D W ) ≥
2, then D W contributes to the Clifford index since h ( C, D W ) = h ( C, ω C ( − D W )) ≥ dim W ≥ . So we have thatdeg( ω C ( − D W )) ≥ h ( C, ω C ( − D W )) −
1) + c ≥ | W | + c. Then it follows that:deg | W | dim | W | ≥ c dim | W | ≥ cg − c − g − − cg − c − . h ( C, D W ) = 1, then as in the previous case we can conclude:deg D W ≤ k W and since k W ≥ k ≥ c , exploiting the monotonicity of the function f :deg | W | dim | W | = 2 g − − deg D W g − − k W ≥ g − − k W g − − k W = f ( k W ) ≥ f ( c ) = 2 g − − cg − c − . Remark 2.16.
Theorem 2.13 above is not a linear stability result for the system | V | unless k ≤ Cliff( C ) and deg | V | = 2 g − − k , i.e. | V | has the biggest possible baselocus, according to Lemma 2.14. Corollary 2.17.
Let V ⊆ H ( C, ω C ) be a vector subspace of codimension k , with k ≤ Cliff( C ) . If deg | V | = 2 g − − k then | V | is linearly semistable. In particular the morphism induced on C is Chowsemistable.Proof. Let W ⊆ V . Let h ≥ k be the codimension of W in H ( C, ω C ). By Lemma 2.13we have that, for m = min { Cliff( C ) , h } ,deg | W | dim | W | ≥ g − − mg − h − m . Now, m ≥ k , and we are done by the monotonicity of the function f defined in (2.15):deg | W | dim | W | ≥ g − − mg − m − f ( m ) ≥ f ( k ) = 2 g − − kg − k − | V | dim | V | . Example 2.18.
Given k ≤ Cliff C points p , . . . , p k on C in general position, clearlythe system | ω C ( − p . . . − p k ) | satisfies the assumptions of Corollary 2.17, asdeg( ω C ( − p . . . − p k )) = 2 g − − k and h ( C, ω C ( − p . . . − p k )) = g − k. Example 2.19.
We see here that indeed for any set of k ≤ Cliff( C ) points on C , thesystem | ω C ( − p . . . − p k ) | satisfies the assumptions of Corollary 2.17. Indeed, we claimthat h ( C, p + . . . + p k ) = 1 . Assume by contradiction that h ( C, p + . . . + p k ) ≥
2: we would have a g d on C with d ≤ k hence gon( C ) ≤ d, but from Ballico’s result (2.10) we obtain: k + 2 ≤ gon( C ) ≤ d ≤ k, which gives a contradiction. From Riemann-Roch theorem h ( C, ω C ( − p . . . − p k )) = 2 g − − k + 1 − g + h ( C, p + . . . + p k ) = g − k. Hence the linear series | ω C ( − p . . . − p k ) | satisfies the hypothesis of Corollary 2.17, soit is linearly semistable. 10 Xiao’s method for subsheaves
Definition 3.1.
We call fibred surface or sometimes simply fibration the data of amorphism f : S → B from a smooth projective surface S to a smooth projective curve B which is surjective with connected fibres.We denote with b = g ( B ) the genus of the base curve. A general fibre F is a smoothcurve and its genus g = g ( F ) is by definition the genus of the fibration. From now on,we consider fibrations of genus g ≥ K f := K S − f ∗ K B (resp. ω f := ω S ⊗ ( f ∗ ω B ) ∨ ) the relative canonical divisor(resp. line bundle). Recall that given a surface S a ( − C ⊆ S such that C = −
1. We say that f is relatively minimal if it doesnot contain any ( − K f being arelatively nef divisor.Throughout the paper, we will assume that f is relatively minimal. Definition 3.2.
We say that a fibred surface is:- smooth or a
Kodaira fibration if every fibre is smooth;- isotrivial if all smooth fibres are mutually isomorphic;- locally trivial if f is smooth and isotrivial (equivalently if f is a fibre bundle):- trivial if S is birationally equivalent to F × B and f corresponds to the projectionon B . If b > f is relatively minimal this is equivalent to S = F × B ,Recall the following relative numerical invariants for fibred surfaces • K f = K S − g − b −
1) the self-intersection of the relative canonical divisor; • χ f := χ ( O S ) − ( g − b −
1) = deg f ∗ ω f the relative Euler characteristic (the lastequality follows from Leray’s spectral sequence); • e f := e ( S ) − e ( B ) e ( F ) = e ( S ) − g − b −
1) the relative topological characteristic(with e ( X ) topological characteristic of X ); • q f := q − b the relative irregularity, with q = h ( S, O S ) irregularity of S .For those invariants the following relations are known [3],[2], [11] (i) K f ≥ K f = 0 if and only if f is locally trivial (see Remark 3.3); (ii) χ f ≥ χ f = 0 if and only if f is locally trivial; (iii) e f ≥ e f = 0 if and only if f is smooth; (iv) q f ≤ g and equality holds if and only if f is trivial.From Groethendieck-Riemann-Roch theorem we have Noether’s relation [1]12 χ f = K f + e f . emark 3.3. Suppose that K f = 0. Then by the slope inequality we have χ f = 0so f is locally trivial. If, on the other hand, f is locally trivial; then by ( ii ) we have χ f = 0, then by Noether’s relation and the non-negativity of e f we have K f = 0. Definition 3.4.
The rank g vector bundle f ∗ ω f is called the Hodge bundle of the fibredsurface.We have the following decompositions of the Hodge bundle as a direct summand ofvector sub-bundles: • (First Fujita decomposition [20]) f ∗ ω f = O ⊕ q f B ⊕ E , (3.5)where E is nef and H ( B, E ∨ ) = 0; • (Second Fujita decomposition [21] [14]) f ∗ ω f = A ⊕ U , (3.6)with A ample and U unitary flat. Definition 3.7.
Following [24], we define the unitary rank u f of the fibred surface tobe the following integer u f := rk U . Remark 3.8.
Comparing the two decomposition, since every trivial bundle is unitaryflat, we have: O ⊕ q f B ⊆ U , and then it holds that q f ≤ u f . Moreover, deg U = 0 and deg A >
0, hence χ f = deg f ∗ ω f = deg A , and u f = g if and only if χ f = 0 (equivalently f is locally trivial). Catanese andDettweiler first gave examples [13] [14] of fibred surfaces for which the unitary summandis not semiample, thus disproving a long standing conjecture of Fujita. They proved thatsemi-ampleness of the Hodge bundle is indeed equivalent to U having finite monodromy.In all the examples in loc. cit. q f = 0, hence in particular the strict inequality q f < u f holds. Note moreover that for any fibred surface such that the monodromy of U isinfinite, the inequality q f < u f also holds “up to base change”, i.e. for any fibration˜ f obtained from f via base change, we still have q ˜ f < u ˜ f . On the other hand, if themonodromy is finite, then there exist a base change a : ˜ B → B such that the inducedfibration ˜ f has q ˜ f = u ˜ f . See [24].Over the moduli space M g of smooth curves of genus g , the function:[ C ] (cid:55)→ Cliff( C )is a well defined lower semicontinuous function This allows us to give the following: Definition 3.9.
Given f : S → B a relatively minimal fibred surface. We define c f := max t ∈ B { Cliff( F t ) | F t is a smooth fibre of f } = Cliff( F ) for F general fibre of f and call it the Clifford index of f . 12 .2 Xiao’s technique
In this section we recall the main results of Xiao’s method, introduced by Xiao in hisseminal paper [40], and further developed by Konno and Barja. We will then applythis method to a subbundle of the Hodge bundle, but we think it is worth to developin full generality and detail the construction, as the precise statement we need is notimmediate to find in the literature. Let π : P B ( E ) → B be the projective bundle ofone dimensional quotients of E (Grothendieck’s notations); and let O P ( E ) (1) be theassociated tautological line bundle. Definition 3.10.
We say that E is a nef (resp. ample) vector bundle if O P ( E ) (1) is nef(resp. ample) over P B ( E ).Let f : S → B be a relatively minimal fibration and fix a divisor D on S . For everynon zero vector subbundle F ⊆ f ∗ O S ( D ), the natural homomorphism f ∗ F (cid:44) → f ∗ f ∗ O S ( D ) −→ O S ( D )yields a rational map S ψ (cid:47) (cid:47) f (cid:31) (cid:31) P B ( F ) π (cid:124) (cid:124) B such that π ◦ ψ = f . The indeterminacy locus of the map ψ is described by the followingresult, whose proof is immediate. Theorem 3.11. [Ohno [33]] In the above situation, there exists a blow up (cid:15) : ˆ S → S and a morphism λ := ψ ◦ (cid:15) : ˆ S → P B ( F ) such that λ ∗ L F ∼ (cid:15) ∗ ( D − Z ) − E where • Z is an effective divisor on S ; • E is a (cid:15) − exceptional effective divisor of ˆ S ; • L F a hyperplane section of P B ( F ) i.e. a divisor associated to O P ( F ) (1) . Definition 3.12.
In this setting we define: • M ( D, F ) := λ ∗ L F the moving part of the vector subbundle F ; • Z ( D, F ) := (cid:15) ∗ Z + E the fixed part of the vector subbundle F ; • N ( D, F ) := M ( D, F ) − λ ∗ µ ( F ) F where we note that (cid:15) do not change the generalfibre of f ; then we can rewrite: N ( D, F ) = M ( D, F ) − µ ( F ) F with F a generalfibre of f .The Xiao’s method makes a crucial use of the Harder-Narasimhan filtration. Definition 3.13.
Let F a vector bundle over a smooth projective curve B . Thereexists a unique sequence of vector sub-bundles of F :0 = F (cid:32) F (cid:32) . . . (cid:32) F k − (cid:32) F k = F (3.14)satisfying the conditions: 13 for i = 1 , . . . , k F i / F i − is a semistable vector bundle; • For any i = 1 , . . . , k setting µ i := µ ( F i / F i − ), we have: µ > µ > . . . > µ k . The filtration 3.14 is called
Harder-Narasimhan filtration of F .We set µ − ( F ) := µ k , and call it the final slope of the sheaf. Remark 3.15.
Note that it holds the formula:deg F = k (cid:88) i =1 r i ( µ i − µ i +1 ) . Indeed, considering the exact sequence of vector bundles:0 → F k − → F k → F k / F k − → , from the additivity property of degree, we can say deg F k = deg F k − + deg F k / F k − . Similarly, we have that: deg F k − = deg F k − +deg F k − / F k − , and so on. By inductionwe can conclude that:deg F k = deg( F k / F k − )+deg( F k − / F k − )+ . . . +deg( F / F )+deg( F ) = k (cid:88) i =1 deg( F i / F i − ) . Now, from the definition of slope, for every i = 1 , .., k we have deg F i / F i − = µ i ( r i − r i − ) , So, setting µ k +1 = 0 and r k +1 = r k , we obtain the desired formuladeg F = deg F k = k (cid:88) i =1 µ i ( r i − r i − ) = k (cid:88) i =1 r i ( µ i − µ i +1 ) . The Xiao’s method is based on the following fundamental result of Miyaoka.
Theorem 3.16 (Miyaoka) . Let F be a locally free sheaf on a projective curve B . Let Σ be the general fibre of π : P C ( F ) → C . The Q -divisor L F − x Σ is nef if and only if x ≤ µ − ( F ) . Remark 3.17.
From Miyaoka’s result we see straightforwardly that µ − ( F ) ≥ F is a nef vector bundle on B . Remark 3.18.
In the case G = f ∗ ω f , it is important to notice that the second tolast subsheaf is precisely the ample part in second Fujita’s decomposition: F l − = A . Indeed, f ∗ ω f is nef, and the subsheaf U = f ∗ ω f / A is a subsheaf of maximal rank in f ∗ ω f with (minimal) degree 0. For the Hodge bundle the last slope µ l is greater orequal to 0 and µ l = 0 if and only if U (cid:54) = 0.We are now ready to expose the heart of Xiao’s method:14 heorem 3.19. (Xiao’s key Lemma [40]) Let f : S → B be a fibred surface. Let D bea divisor on S and suppose that there exist a sequence of effective divisors: Z ≥ Z ≥ . . . ≥ Z s ≥ Z s +1 := 0 and a sequence of rational numbers µ > µ > . . . . . . > µ s ≥ µ s +1 := 0 such that for every i = 1 , . . . , s N i := D − Z i − µ i F is a nef Q − divisor. Then for anyset of indexes { j , . . . , j s } ⊆ { , . . . , l } we have D ≥ s (cid:88) i =1 ( d j i + d j i +1 )( µ j i − µ j i +1 ) where d j := N j F .Proof. Just observe that the assumptions imply the following: N j i +1 − N j i = ( N j i +1 + N j i )( N j i +1 − N j i ) = ( N j i +1 + N j i )( Z j i − Z j i +1 − ( µ i − µ i +1 ) F ) ≥ ( d j i + d j i +1 )( µ i − µ i +1 ) , and that s (cid:88) i =1 ( N j i +1 − N j i ) = − N j + N j s ≤ N j s ≤ D . We are now ready to state the version of Xiao’s basic result in the form needed. Notethat this is an expanded version of the inequality stated in [10, Remark 24].
Theorem 3.20.
Let f : S → B be a fibred surface. Let D be a nef divisor on S and G ⊆ f ∗ O S ( D ) be a rank r subsheaf. Let d (cid:48) = M F where M = M ( D, G ) .Suppose that there exists a real number α > such that for every linear subsystem | P | of | M | F | deg | P | dim | P | ≥ α. (3.21)(i) The following inequality holds: D ≥ α ( r − r deg G = 2 α ( r − µ ( G ) . (3.22)(ii) If moreover G is nef, then, for every non negative integer d ≤ d (cid:48) , the followinginequality holds: D ≥ αdd + α deg G . (3.23)15 roof. Let 0 (cid:32) G (cid:32) . . . . (cid:32) G k − (cid:32) G k = G (3.24)be the Harder-Narasimhan filtration of G . We note that in general this filtration neednot necessarily be related to the Harder-Narasimhan filtration of f ∗ O S ( D ) (althoughthis will happen in the application: see Remark 4.4).Following Ohno’s construction in Theorem 3.11, we consider a suitable blow up ν : ˆ S → S and over ˆ S for every index i we consider the divisors M i := M ( D, G i ) and Z i := Z ( D, G i ), which are respectively nef and effective. Call r i = rk G i and d i := M i F .We also set G k +1 := G k = G . Let us first assume that G s nef and prove inequality (3.20). The final slope of G is µ k ≥ µ k +1 = 0 and Z k = Z k +1 . The sequence( Z i , µ i ) clearly satisfies by construction: Z ≥ Z ≥ . . . ≥ Z k = Z k +1 , and µ > µ > . . . > µ k ≥ µ k +1 := 0 . Observing that µ i coincides with µ − ( G i ) we have by Miyaoka’s Theorem 3.16 that thedivisors N i := M ( D, G i ) − µ i F are all nef Q − divisors over ˆ S . Since the intersection product is invariant under bira-tional morphism we have ( ν ∗ D ) = D . So, we can apply Theorem 3.19 to estimate( ν ∗ D − Z k ) . We make a wise use of the choice of the indexes in the theorem.Firstly we use the set of indexes { , . . . , k } , obtaining the inequality( ν ∗ D − Z k ) ≥ k (cid:88) i =1 ( d i + d i +1 )( µ i − µ i +1 ) , which in its extensive form reads as follows( ν ∗ D − Z k ) ≥ ( d + d )( µ − µ ) + . . . + ( d k − + d k )( µ k − − µ k ) + ( d k + d k +1 )( µ k ) . Observe that assumption (3.21) implies that for any i , d i ≥ α ( r i − r = 1, the inequality holds trivially. Using this inequality and the fact that r i ≥ r i − +1for i = 1 , . . . , k − r k +1 = r k , we have:( ν ∗ D − Z k ) ≥ k (cid:88) i =1 ( d i + d i +1 )( µ i − µ i +1 ) ≥≥ α ( k − (cid:88) i =1 r i ( µ i − µ i +1 ) + r k µ k ) − α ( µ + µ k ) == 2 α deg G − α ( µ + µ k ) . Consider now the list of indexes { , k } : we have( ν ∗ D − Z k ) ≥ ( d + d k )( µ − µ k ) + ( d k + d k +1 )( µ k ) ≥ d k ( µ + µ k ) . ν ∗ D − Z k ) ≥ αd k d k + α deg G . Now observe that ( ν ∗ D − Z k ) = D − ν ∗ DZ k + Z k ≤ D , where the last inequality follows form the fact that ν ∗ D is nef and Z k effective andfrom Z k ≤ g ( t ) := 2 αtα + t , which is monotone increasing for t ≥
0. From the hypothesis we have d k ≥ d so we candeduce that D ≥ αd k d k + α deg G = g ( d k ) deg G ≥ g ( d ) deg G = 2 αdd + α deg G , and the proof of inequality (3.20) is concluded under the assumption that G is nef.In the non-nef case, just consider as in [10, Prop.8] the last nef subbundle in theHarder-Narasimhan sequence: G s , where s = max { i | µ i ≥ } . Applying the very sameconstruction to G s we can obtain D ≥ αd s d s + α deg G s ≥ α ( r s − r s deg G s ≥ α ( r − r deg G , where the second inequality is obtained by choosing d = α ( r s − g above and from the fact that clearlydeg G s ≥ deg G . So, also inequality (3.22) is proved. Remark 3.25.
As proved in [6], the vector subbundle G s in the proof of the abovetheorem is a maximal element in the set of nef sub-bundles of G : for any nef sub-bundleof G it holds F ⊆ G s .In particular, if |G ⊗ C ( t ) | is linearly semistable for general t ∈ B , we can take: α = deg |G ⊗ C ( t ) | dim |G ⊗ C ( t ) | . and obtain the following well known result (see [10]). Corollary 3.26.
Let f : S → B be a fibred surface. Given D a nef divisor on S and G ⊆ f ∗ O S ( D ) a rank r subsheaf. Let d = deg |G ⊗ C ( t ) | the degree of the linear system |G ⊗ C ( t ) | , over a general fibre F t . If |G ⊗ C ( t ) | is linearly semistable, then D ≥ dr deg G = 2 dµ ( G ) . Slope inequalities
Let f : S → B be a relative minimal fibration of genus g ≥
2. We are now ready toprove our main estimates on the slope of fibred surfaces.Firstly, using the first Fujita decomposition (3.5) we give a bound that improvesthe main bound of [8]. Note that the proof is much simpler than the proof of [8], wherewe needed to lift a general projection on the fibre to obtain the desired subsheaf of theHodge bundle.
Theorem 4.1.
Let m := min { q f , c f } . The following inequality holds: K f ≥ g − − mg − m χ f . Proof.
First observe that in the hyperelliptic case m = 0 and the inequalities are justthe classical slope inequality. Assume that the general fibre is not hyperelliptic.Let us consider the first Fujita decomposition (3.5). f ∗ ω f = E ⊕ O ⊕ q f . If q f ≤ c f consider the vector bundle G := E . If q f ≥ c f consider the vector bundle G := E ⊕ O q f − c f B In both cases the fibre over a general t ∈ B G ⊗ C ( t ) ⊆ H ( F t , K F t )defines a linear subsystem of H ( F t , K F t ) of codimension m .Let us start by observing that in case that the first vector subbundle in the Harder-Narasimhan filtration of the Hodge bundle is of rank one (a line bundle), we have d = 0 = r −
1. By the remark above and Theorem 2.13, we can apply Theorem 3.20to D = K f and G as defined above, with α = g − − mg − m − . We thus obtain K f ≥ αdα + d deg G = 2 2 g − − mg − m χ f , as desired.We shall now turn our attention on the influence of the unitary rank u f on theslope. Theorem 4.2.
The following inequalities holds: K f ≥ (2 g − − u f )( g − u f ) χ f u f ≤ c f (2 g − − c f )( g − u f − g − c f − g − u f ) χ f u f ≥ c f Proof.
As above, we can assume thet F is non-hyperelliptic. Consider the second Fujitadecomposition (3.6) f ∗ ω f = A⊕U . As already observed, we have that deg A = deg f ∗ ω f .We distinguish the two following cases: • If u f ≤ c f , then consider G = A From Theorem 3.14 we can estimate the degreeof that linear subsystem as follows:deg |A ⊗ C ( t ) | ≥ g − − mg − m − g − u f −
1) = 2 g − − u f =: d. D = K f and G = A , we have: K f ≥ αdα + d deg A = 2 2 g − − u f g − u f χ f , as wanted. • If u f ≥ c f , using Theorem 2.13 we estimate the degree of the linear system |A ⊗ C ( t ) | as: deg |A ⊗ C ( t ) | ≥ g − − c f g − c f − g − u f −
1) =: d. Then applying Theorem 3.20 with D = K f and G = A we have: K f ≥ αdα + d deg A = 2 (2 g − − c f )( g − u f − g − u f )( g − c f − χ f , and the proof is concluded. Remark 4.3.
Observe that these last inequalities are not symmetric in min { u f , c f } as the one of Theorem 4.1. In case there exists a unitary flat subsheaf U (cid:48) of U , withrk U (cid:48) ≥ u f − c f , one can improve the last inequality of Theorem 4.2. However, such asubsheaf U (cid:48) need not to exist. Remark 4.4.
It is worth making the following remark. In Xiao’s method as exposedin Section 3.2, we use the Harder-Narasimhan sequence of the subsheaf G of f ∗ O S ( D ).This in general is not related to the Harder-Narasimhan sequence of f ∗ O S ( D ) itself.But in case G is a nef subsheaf of the Hodge bundle that contains the ample summand A , then the Harder-Narasimhan filtration of G clearly is the truncation of the filtrationof f ∗ ω f . Now we want to expose a fist example of a fibred surface in which the bound of Theorem4.2 is better than the bound of Theorem 4.1 and of any other previous bound. Theknown examples of fibred surfaces with high unitary rank ([13, 14, 15, 29]) all satisfy c f ≤
1. It is therefore interesting to find examples with Clifford index close to theunitary rank. This is a first example in this direction.We use the same construction of [15], although with some modifications, and referto loc. cit. for details. Consider a family of cyclic coverings of P of degree 7, givenbirationally by the equation: y = x x ( x − x ) ( x − tx ) , (5.1)where x , x are homogeneous coordinates of P , t ∈ C \ { , } and we can think y asa section of O P × P (1 , S : → B with the following properties:19 the base curve B is of genus b = 3; • the general fibre F is a smooth curve of genus g = 6; • there are only three singular fibres, which are given by two smooth curves of genus3 intersecting transversely (in particular f is a semistable fibration); • on every fibre there is an action of Z / Z by automorphisms; • the irregularity q = h ( S, O S ) is equal to 3.This last property allows us to say that f : S → B is an Albanese fibration, i.e. that q f = 0.The action of Z / Z on the smooth fibres the space induces an action on H ( F, O F ) = H ( F, ω F ) and for any character χ j ∈ Z / Z we can calculate the dimension of thecorresponding characteristic subspace H ( F, ω F ) j via the Chevalley-Weil formula:dim H ( F, ω F ) j = − (cid:88) k =1 ( − jα k ) j = 11 if j = 2 , , ,
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